From Schaum's Outline of Vector Analysis: – Chapter 2. Required ... (c) Suppose
that all four points lie on the surface of a sphere S. De- termine the center point ...
Math 601 Homework 8 Due Friday, October 26 Solutions should be typed or written neatly and legibly. Answers should be explained. You should reference all your sources, including your collaborators. For more information on writing up homework solutions, see the guidelines at the beginning of Homework 1. Reading assignment: • From Linear Algebra and Vector Calculus at Texas A&M : – Sections 1.4–1.5, 1.7, 5.1–5.3 • From Schaum’s Outline of Vector Analysis: – Chapter 2 Required problems. Turn in a solution for each of the following problems. 1. Find the distance between the following two lines: 1 0 x y = 1 + t 1 0 2 z x 1 1 y = −1 + t 1 z 3 0 2. Consider the four points P1 = (5, 2, 0), P2 = (1, 5, −1), P3 = (1, 7, 3), and P4 = (4, 6, 3) in R3 . (a) Find the equation for the plane consisting of all points equidistant from P1 and P2 . (b) Do the same for P1 and P3 and for P1 and P4 . (c) Suppose that all four points lie on the surface of a sphere S. Determine the center point of S.
3. Consider the following system of partial differential equations: −xy 2 ∂u = ∂x (x2 + y 2 )2
x2 y ∂u = and ∂y (x2 + y 2 )2 ∂u ∂u and in terms (a) Given the above system, find equations for ∂r ∂θ of the polar coordinates r and θ. (b) Solve the system of equations that you obtained in part (a). 4. Consider the coordinates u, v on R2 defined by the equations: 1 x = uv and y = (u2 − v 2 ) 2 (a) Draw a sketch of the plane showing the gridlines u = −2, u = −1, u = 1, and u = 2, and v = −2, v = −1, v = 0, and v = 2. ∂x ∂y ∂x ∂y (b) Determine , , , and . Express your answers in terms ∂u ∂u ∂v ∂v of u and v. ∂u ∂u ∂v ∂v (c) Determine , , , and . Express your answers in terms ∂x ∂y ∂x ∂y of u, v, and y. ZZ 2y dA, where D is the region bounded by 5. Evaluate the integral y = x and y = x2 .
D
6. Compute the following integral: Z eZ
1
sin(y 2 ) dydx x ln x 1 (Hint: Reverse the order of integration.) Recommended problems. It is recommended that you do many more problems than the required problems. The following list of problems are good practice problems. • From Linear Algebra and Vector Calculus at Texas A&M : – – – – – –
Section Section Section Section Section Section
1.4: 1.5: 1.7: 5.1: 5.2: 5.3:
# # # # # #
5–21 odd 1–27 odd 1–35 odd 1, 3, 5, 9, 11, 13 1–15 odd, 21, 23, 25 1–13 odd
• From Schaum’s Outline of Vector Analysis: – Chapter 2: # 18, 19, 28, 29, 31, 32, 39, 45, 65, 67, 77–79, 82–84, 90, 95–97
